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Suplemento de la Revista Latinoamericana de Metalurgia y Materiales2009; S1(1): 223-234

0255-6952 2009 Universidad Simn Bolvar (Venezuela) 221

DESIGN OF A DIE FOR THE COLD COMPACTION CALIBRATION OF POWDERED

MATERIALS

Mauricio Barrera*, Hctor Snchez S.

Este artculo forma parte del Volumen Suplemento S1 de la Revista Latinoamericana de Metalurgia y Materiales(RLMM). Los suplementos de la RLMM son nmeros especiales de la revista dedicados a publicar memorias decongresos.

Este suplemento constituye las memorias del congreso X Iberoamericano de Metalurgia y Materiales (XIBEROMET) celebrado en Cartagena, Colombia, del 13 al 17 de Octubre de 2008.

La seleccin y arbitraje de los trabajos que aparecen en este suplemento fue responsabilidad del ComitOrganizador del X IBEROMET, quien nombr una comisin ad-hoc para este fin (vase editorial de estesuplemento).

La RLMMno someti estos artculos al proceso regular de arbitraje que utiliza la revista para los nmeros regularesde la misma.

Se recomend el uso de las Instrucciones para Autores establecidas por la RLMM para la elaboracin de losartculos. No obstante, la revisin principal del formato de los artculos que aparecen en este suplemento fueresponsabilidad del Comit Organizador del X IBEROMET.

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Suplemento de la Revista Latinoamericana de Metalurgia y Materiales2009; S1(1): 223-234

0255-6952 2009 Universidad Simn Bolvar (Venezuela) 223

DESIGN OF A DIE FOR THE COLD COMPACTION CALIBRATION OF POWDERED

MATERIALS

Mauricio Barrera*, Hctor Snchez S.

Escuela de Ingeniera de Materiales, Universidad del Valle. Cali, Colombia

E-mail: [email protected]

Trabajos presentados en el X CONGRESO IBEROAMERICANO DE METALURGIA Y MATERIALESIBEROMET

Cartagena de Indias (Colombia), 13 al 17 de Octubre de 2008

Seleccin de trabajos a cargo de los organizadores del evento

Publicado On-Line el 20-Jul-2009Disponible en: www.polimeros.labb.usb.ve/RLMM/home.html

Abstract

This paper addresses the problem of finding the radial stress within a powder body undergoing plastic deformation by

confined compression, motivated by the compaction test for powdered materials. Small deformation, elastoplastic

behaviour is assumed, with porosity as the parameter governing the evolution of material properties. An elliptic yieldsurface is selected, whose aspect ratio approaches zero as full density is neared, so as to recover the Von-Mises criterion of

plastic yield in ductile metals. The results were used to track the radial stresses on a notional compaction process, and

applied to the design of an instrumented closed die. In addition, the aspect ratio of the yield surface showed an evolution

according to the widely used Heckel porosity-pressure function [6], thus extending the interpretation of this model to amore general stress state.

Keywords:Powder compaction, Powder plasticity, Plasticity model calibration

Resumen

Este articulo trata el problema de hallar el esfuerzo radial dentro de un espcimen de material en polvo sometido a

deformacin plstica mediante compresin confinada, motivado por el ensayo de compactacin para materiales en aquel

estado. Se asume un comportamiento elstico-plstico de pequeas deformaciones, siendo la porosidad el parmetro quegobierna la evolucin de las propiedades del material. Se escoge una superficie elptica de fluencia plstica, cuya relacin

de aspecto se aproxima a cero en la medida que el material se acerca a la densificacin total. De este modo se recupera elcriterio de fluencia plstica de Von-Mises. Los resultados del anlisis se usaron para calcular los esfuerzos radiales duranteun proceso de compactacin propuesto para el caso, y se aplicaron al diseo de un dado de compactacin instrumentado.

Adicionalmente, la relacin de aspecto de la superficie de fluencia mostr una evolucin de acuerdo con el modelo

porosidad-presin de Heckel [6], extendiendo as la interpretacin que se hace de este modelo a una situacin de estado deesfuerzo mas general.

Palabras clave:Compactacin de polvos, Plasticidad de polvos, Calibracin de modelos de plasticidad

1 INTRODUCTION

The current context within which manufacturing

processes of powder materials are engineered

demands a mechanical characterization of thematerial, considering the multiaxial nature of thestress-strain state. To this purpose the conventional

close compaction test [1] must allow for computing

the radial stress into the specimen, in addition to the

punching -or axial- stress. In particular, a set of

strain gauges measuring the hoop strain at the outer

surface of the die is laid out to fulfil the requirement

[2,3]. Having recourse to thick walled vessel

formulae [4] leads to a prompt calculation of the

radial stress present in the powder specimen. A

crucial point is the determination of an appropriate

value for die wall thickness: too thin and it would

collapse upon loading of the powder specimen; too

thick and hoop strains could not be properly gauged.

In the light of this, it is clear that some estimation of

radial stress exerted by the powder body, which is

undergoing uniaxial confined compression, is to be

done in advance. In a first approximation to this

problem the analysis of seemingly influential

factors, such as die-wall friction and

nonuniformities within the specimen thereof -both

in axial and radial directions-, is relinquished on

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Barrera et al.

224 Rev. LatinAm. Metal. Mater.2009; S1(1): 223-234

behalf of mathematical tractability. That said, it was

decided to turn to a yield surface expression devised

for aluminium foams [5], deeming it as makeshift

powdered material. The end result is a solution to

the problem of uniaxial confined compression of a

specimen of increasing relative density, where

elastic stresses are stored up until plastic onset,dictated by the mentioned material model, occurs.

In this regards, and to completely define the

problem an elastoplastic behaviour assumption

comes necessary, whereby elastic behaviour is

assumed to be linear and of small strains, and also

showing a dependence on the degree of compaction.

Elastic computations provide the stress state by

means of which plastic yielding is arrived to; the

yield function and an assumed associative plastic

behaviour provide an update of plastic straining and,

hence, of porosity. This is then used to compute new

values of elastic constants. The application of thisprocedure throughout a compaction process

permitted to track the evolution of radial stresses,

and therefore, the application of thick wall vessel

formulae to make a decision in regards of die wall

thickness. The compaction die was built and the

requirements of structural strength and hoop strain

measurement were satisfactorily met. The evolution

of the yield surface parameter, called aspect ratio,

showed to obey a Heckel-like function [6], thus

providing an interpretation of this porosity-pressure

model in terms of a more general stress state.

2 THE YIELD FUNCTION

The classic Von-Mises yield function has been

modified by several authors intending to represent

the observed behaviour in porous materials, where a

ductile, metal-like mechanical deformation at full

density is attained eventually. One such model is

given by Eq.(1):

( ) 00 = kg ij (1)

The first term represents a function of stress

components which is the equivalent stress fromVon-Mises theory, including a modification by

hydrostatic pressure. The second term stands for

yield strength in uniaxial test, whether tension or

compression. It shows a dependency on the extent of

plastic deformation through an observable

parameterk. The function of stresses is given byEq.(2) [7]:

( )( )

+

+=

2

222

31

meijg (2)

It is worth to note how this function is actually a

quadratic expression in{ }me , depicting an ellipseon ( )qp, plane [9], which has proven adequate infitting experimental data to results of triaxial test

probing of metal foams and some granular

materials. It makes part of a more general family of

elliptic yield functions of the form [8]:

0' 202

12 =+ JJ (3)

posed in terms of stress invariants ( )', 21 JJ forstress tensor and stress deviator, and two empirical,

material dependent parameters ( ), .The stress components shown in the numerator of

Eq.(2) stand for the square root of the double

contraction of stress deviator equivalent or

effective stress- and one third of the trace of

hydrostatic component of stress tensor-mean stress-,

as set out in Eq.(4) and Eq.(5):

21

2

3

= ijije SS

(4)

kkm 31=

(5)

The parameter captures the dependency of yieldfunction on hydrostatic stress. At full density,

0= and the yield function of Von-Mises isrecovered. In deriving the yield function, intended to

characterize aluminium foams, a material specimen

was probed using a triaxial load cell, and the results

of Figure 1 were obtained.

It is shown that the yielding of the material is

described by an ellipse lying on the ( )qp, planecharacterized by the ratio of its semi axes. As thisis a material-dependent quantity, the parameter isa material property called pressure-sensitivity

coefficient.

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Design of a die for the cold compaction calibration of powdered materials

Rev. LatinAm. Metal. Mater.2009; S1(1): 223-234 225

Figure 1. Experimental yield loci for some typical

aluminium foams and comparison with elliptic yield

surface [7].

As to the coordinate axes in Figure 1, Figure 2recalls the stress state for a material element

undergoing closed die compaction, in terms of

punching pressure and radial pressure. Stress

components homogeneity throughout the specimen

is assumed.

axial

Vaxial =

Hradial =

Figure 2. Stress state of a homogeneous specimen

undergoing confined compression.

Certainly a relationship between equivalent and

mean stress with p and q components of stress may

be established, so that Eq.(6) and Eq.(7) hold [9]:

( )HVe q == (6)

3

2 HVm p

+== (7)

3 CONFINED COMPRESSION OF A

METAL FOAM

The aim is to know what the radial pressure is that a

specimen of metal foam, undergoing closed die

compaction, would exert. This material is

considered, in the approximation presented here, as

appropriately taking the place of a metal powder

body. Replacing Eq.(2) in Eq.(1) and from the

considerations leading to Eq.(6) and Eq.(7) it is

possible to state Eq.(8)

( )( )( )

( )

( )

+

+

=HVm

HVek

,

,

31

122

2

20

(8)

The observable material parameter kto the left handside of the equation has been made dependent on

density, i.e. density is the quantity used to track the

evolution of plastic deformation of the material.

As the punching stress, represented by V , is the

experimental input in a closed die compaction test,

and also as this happens to be the yield strength ofthe tested material throughout compaction, the aim

is to solve Eq.(8) for it. Nonetheless an issue is yet

to be addressed, namely, the way the radial stress

exerted against the die wall is generated. At thispoint it is assumed that elastic behaviour is

responsible for storing that energy which carries

over to exerted radial pressure. Linear elastic, small

strains behaviour is assumed at this point. This in

conjunction with presumed homogeneity within thespecimen permits to state the generalized Hookes

law to an axisymmetric element, as shown in Figure

3 and rearranged in Eq.(9).

( )[

( )[ ]

( )[ ]

+=

+=

+=

PE

PE

PE

rr

rr

rrzz

10

10

1

zz

rr

rr

zz

rz

zr

Figure 3. Axisymmetric stress state and generalized

Hooke law to the case.

=

B

A

A

P

AA

BA

AB

zz

rr

1

0

0

(9)

In Eq.(9)1 EA and 1EB . The linear

dependence of the first two equations in Eq.(9)

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shows that radial and transverse stress components

equal each other. This is why a single component is

given to them in the context of stress states on

( )qp, plane, where axisymmetric stress states areassumed (i.e. Hrr == ).

Solving for radial stress yields Eq.(10):

PKAB

APrr =

= (10)

The last term to the right is just shorthand for

upcoming derivations. This result may now be

replaced back into Eq.(8) and solved for punching

stress PVzz == , resulting in Eq.(11) below:

( )

( )( ) ( )

21

222

2

0

219

19

1

++

+

=KK

P

(11)

Eq.(11) can be thought of as the yield strength of the

material under closed die compaction. It is worth torecall that a yielding point is arrived to by means of

storing energy by previous deformation, and it is the

elastic regime of the material where this storage

takes place.

So, here lies the central assumption to the derivation

just conducted: the radial stress component is taken

to be an immediate consequence of the confined

compaction carried out by the punching stress and a

presumed linear elastic behaviour of the material. It

is this link between stress components, stemming

from the axisymmetric, confined compression

mechanical condition what allowed for the

expression in Eq.(11) to be gained.

4 EVOLUTION EQUATIONS: YIELD

SURFACE

It is left now to specify the way in which the yield

surface aspect ratio , yield strength under uniaxial

loading ( )( ) k0 and elastic properties Kdependupon a measure of plastic deformation. Relativedensity will serve the purpose, in a first

approximation since, in so doing, the second

invariant of the strain tensor is being neglected as a

measure of deformation. Then ( ) and( ) ( ){ } ,E shall be sought in the sequel.

As to the yield surface aspect ratio, it is necessary to

recall that Eq.(2) was originally intended to

represent the yield behaviour of a metal foam. Such

a material is not purported to deform plastically by

reducing its porosity up until attaining full density,

unlike metal powders during compaction. The yield

function in Eq.(2), within the context of metal

foams, is said to evolve in a self-similar manner, so

that the aspect ratio is virtually a constantthroughout the plastic deformation process. The

hardening rules to that particular case are derived

from application of Prandtl-Reuss plastic flow rule

and definition of a work conjugate stress-strain pair,

in such a way that the tangent modulus from readily

conducted tests, such as hydrostatic and uniaxial

loading, may be used to characterize the way a

material hardens plastically.

To make the case for powder compaction one could

resort to some results of triaxial probing over metal

powder specimens showing that, at least during the

first stages of compaction, the yield surface is

appropriately represented by an ellipse. Now, as the

powder is intended to attain full density, aspect ratio

must be made variable with relative density. So, an

evolution like that set out in Figure 4 is pursued.

q

p02P01P

02Q

01Q

2

1

iQ0 i

iP0

fullsurfaceyieldMisesVon

0

0

P

Q

Figure 4. Intended yield surface and its evolution with

relative density.

A hint to the functional form of ( ) is provided bythe measured evolution of the Cam-Clay yield

function as applied to some metal powders of

industrial interest. The evolution of the ellipses,

characterized by their semi axis lengths ( )00 ,QP isdescribed by the function shown in Eq.(12) [2].

=

max

01max0 tanh

Q

PKQQ (12)

It is clearly a non trivial task to solve analytically

for1

00

= PQ . Yet, from the same reference, 0P istaken as the yield surface parameter uniquely

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defining yield surface shape and also uniquely

depending on relative density. A particular form of

this dependency was found to be given by Eq.(13)

3

0

20

1ln

K

full

KP

=

(13)

In this equation ( )max0 ,, stand for currentdensity, apparent density and maximum (theoretical)

density of the powdered material, respectively.

Equations Eq(12) and Eq.(13) turned out to be

satisfactory fitting functions and are not

prescriptive, i.e. experimental data from other

materials may quite well be fitted by some other

functions. Nonetheless, it is always preferred to use

functions which are somehow related with powder

compaction models, particularly in regards ofEq.(13). Being this the case, a physical meaning of

the fitting parameters is gained. In this sense it is

worth to mention that compaction models relating

0P with , such as that of Heckel [6], are widely

employed for this specific purpose.

The case was made for an iron alloy powder, which

bore a yield strength of 170MPa ( maxQ ) after

compaction. Fitting parameters were 1.5, 21.903 and

1.3527 for { }321 ,, KKK respectively [2]. Tabulatingdata from Eq.(12) and Eq.(13) provides a mean to

track the progress ( ) sought for.

Table 1 displays the results of a compaction process

of up to 87.5% of full density for an iron powder

(density 7.8Kg/m3), which is what in practice is

reached to such a material.

Table 1.Data from a compaction process, iron powder.

Current

density

(Kg/m3)

P0(MPa) Q0(MPa)Aspect

Ratio

2.808 0.000 0.000 -

5.803 19.460 28.907 1.485

6.802 41.694 59.864 1.436

7.301 67.682 90.958 1.344

Apparent density (Kg/m3): 2.808; Theoretical

density (Kg/m3): 7.8; Qmax(MPa): 170 [2]

It can be seen that the aspect ratio changes are

negligible and that a Von-Mises yield surface might

not be recovered this way. However, assuming that

from this compaction state up to full density there

are not changes in material phenomena underlying

fitting parameters{ }321 ,, KKK , one may actuallyapproach the theoretical density (7.8 Kgm-3). Resultsare set out in Table 2 below.

Table 2.Yield surface aspect ratio on using Eq.(13) for

extrapolation.

Current

density

(Kg/m3)

P0(MPa) Q0(MPa)Aspect

Ratio

7.650 119.603 133.257 1.114

7.794 292.428 168.060 0.575

7.797 337.976 169.129 0.500

7.799 429.565 169.827 0.395

It can be seen that the yield surface actually tends to

approach Von-Mises yield surface in a noticeablemanner only as density is within 2% before

theoretical density. Then, one may take these just

tabulated results to perform a curve fitting leading to

an evolution equation ( ) . The following variablemodification, Eq.(14), is enforced so that curve

fitting results do not resort to an unwieldy set of

empirical constants, which would make a physical

interpretation nearly impossible.

( )

=

full

LnLn

1

1 (14)

Plotting this equation leads to Figure 5 below.

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

( )Ln

( )

full

Ln1

1

Figure 5.Evolution of yield surface aspect ratio with a

logarithmic function of relative density.

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From the value 2.00 (corresponding to a relative

density of about 87%) for the abscissa towards the

right, a straight line would provide a satisfactory

fitting, indicating an exponential-like relationship

between the plotted parameters. So, the

dependence ( ) could be split into two ranges:

From apparent density to 87% full density: aspect

ratio is almost a constant, with a value of 5.1 .Yield surface evolution is self similar, and provided

a triaxial probing over a body of metal powder

confirms this self similarity, metal foam yield flow

behaviour might be thought of as a fairly good

imitation of that of metal powder. This does not

mean that at a micro-scale level the same plastic

flow mechanisms come upon in metal foams and

metal powders.

From relative density 87.5% up until full, theoretical

density: aspect ratio varies in order to recover theVon-Mises yield surface as the material gains full

density. It is by virtue of Eq.(13) that self similarity

breakdown takes place. As full density is neared 0P

tends to infinity and 0Q tends to maxQ . As seen, this

could only be explored by means of curve fitting

functions, for the practical limits of powder

compaction processes prevent compaction densities

close to 100% from being achieved.

Cold compaction of a typical iron powder shows, in

addition, that most of the compaction takes place at

relative densities above 80%, and indicates apractical limit of about 7.2 Kg/m3[10].

For relative densities of 80% and above, Figure 6

shows a point selection fitting a straight line. The

fitting function is given by Eq.(15) thereafter.

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0. 00 1. 00 2. 00 3. 00 4. 00 5.00 6. 00 7. 00 8. 00 9. 00 10. 00

( )Ln

( )

full

Ln1

1

Figure 6. Linear fitting to compaction data at high

relative densities (>80%)

( ) ( ) 378.011ln177.0ln +

= full

(15)

Eq.(15) is actually a linearization of Eq.(16), which

is the sought evolution equation ( ) .

( ) ( )177.0

11459.1

= full

(16)

Now, the way in which uniaxial yield strength

0 depends on density is also necessary to perform

computations based on the formulations above, andin particular, to make use of Eq.(11). From metal

foam studies, whose specimens are appropriately

subject to uniaxial loading tests, the following

empirical scaling formula is brought in [5,11],

Eq.(17).

( )2

00 3.0

==full

y (17)

Factors 0.3 and 2 come from curve-fitting exercises,

and are used to sort out different metal foam

microstructures; y stands for the yield strength of

the full dense material. Yet, it is noted that this

scaling law embodies the fact that metal foams are

not intended to deform plastically so as to attain full

density: for a relative density of 1.0 yield strength

would only be one third that of the full densematerial. On accounts of that a much simpler

expression, yet more abiding to experimental

observations, is put forward to use, Eq.(18):

=full

y 0 (18)

Microstructure issues of apparent relevance to metal

powders are spared from analysis. Worth to mention

is that a full dense material obtained from powdercompaction may show plastic deformation

mechanisms different from those of wroughtmaterials: contact points between particles are

usually tainted with impurities which might quite

well influence dislocation slip mechanisms.

5 SCALING EQUATIONS FOR ELASTIC

PARAMETERS

In a like manner to uniaxial yield strength

dependence on density, scaling of elastic parameters

is also brought from rather empirical, curve fitting

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exercises which, however, have proven useful in

leading to handy, yet accurate models of material

behaviour. So, elastic modulus may be related to

density as shown in Eq.(19) [5,11]:

( )

2

3.0 == fullfullEEE (19)

Once again, 0.3 and 2 account for microstructure

features of some specific metal foam; also,

fullE stands for Young modulus to the full dense

material. Poisson ratio has been said to be 0.3,

typical of many full dense structural metals, and not

bearing a noticeable dependence on density.

Nonetheless, an empirical study performed over

porous structural metals obtained by sintering of

porous compacts provided the following

relationship, employed herein [11]:

( )

==full

37.1exp068.0 (20)

6 RADIAL STRESS COMPUTATIONS

From the discussion following Eq.(11) radial stress

computation comes after two steps: First, define a

set of relative densities spanning over the range of

densities. 1...,,, 210 fullfullfull ; Second,

compute the value of material properties (elastic and

plastic) from evolution equations provided. To eachone of the set of relative densities, compute the

punching pressure P from Eq.(11). This punchingpressure is at equilibrium with elastic deformation

of a porous body of the selected relative density, and

imposes a plastic deformation corresponding to that

relative density as well. Finally, radial stress are

computed following Eq.(10).

From elaborated material data and evolution laws,

corresponding to a quite general class of an iron

alloy porous body, computations are shown in Table

3 (the following values were used:

MPaGPaE yfull 200;200 == ).

Table 3.Material data from compaction of an iron alloy powder [UWS, 2007], and resulting radial stresses.

Relative

densityE y P (MPa)

PFullDense

(MPa)

VonMises

Fraction

Radial

Stress

(MPa)

0.448 1.20E+10 0.13 1.500 8.95E+07 93.41 233.52 0.40 13.410.675 2.73E+10 0.17 1.500 1.35E+08 142.07 252.18 0.56 29.40

0.870 4.54E+10 0.22 1.017 1.74E+08 206.46 281.12 0.73 59.58

1.000 6.00E+10 0.27 0.003 2.00E+08 315.15 315.15 1.00 115.15

In addition to the sought figures for radial stress,

data labelled as P Full Dense and Von Mises

Fraction is provided too. The former refers to the

punching load that should be applied if a full dense

material of corresponding yield strength were to be

taken to its plastic yielding onset. Comparison with

the actual punching pressure P required to incurplastic yield in the porous material provides a sense

of how porosity makes for easier yielding under

applied load, and also the way a full dense, Von-

Mises behaviour is recovered as, indeed, full density

is attained and yield surface aspect ratio reducesto zero. The latter provides a figure of this

recovering process, and is plotted in Figure 7 below.

Point 1 in Figure 7 corresponds to the last value

where a constant aspect ratio of 1.5 is employed.Point 2 corresponds to a relative density of 0.87, and

from this point on the evolution equation Eq.(16) is

employed. There is a noticeable change when

moving on between these ranges, due to the fact of

using models which try to represent data in Table 1

and Table 2. Also the sharp transition from point 3to point 4 is a reflection of how the material model

rushes to recover Von Mises plasticity only at

relative densities close to unity.

Radial stress data are used to estimate die wall

thickness in accordance with the requirements laid

down in the Introduction to this paper. This andother details of the closed die design are exposed

next on.

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0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.10

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Relative Density

VonMise

sFraction

1

2

4

3

Figure 7.Approximation to Von Mises-like behaviour at

relative densities close to theoretical.

7 CLOSED DIE DESIGN

Common constitutive equations for mechanicalbehaviour of granular bodies imply a joint evolution

of axial stress, radial stress and porosity within the

specimen, which must be traced out so that the

resulting plots convey the desired material

parameters. This is actually the problem whose

solution is sought when designing a parameter

identification rig. It will be henceforth regarded as

Measuring Instrument, or just MI.

The MI aimed at in this study is as set out below,

Figure 8.

P

Upper punch

Rigid plate

Load cell

Die wall

Powder

Foil strain gage

Figure 8.Scheme of the instrumented closed die.

The upper punch load conveys an axial

stress axial

to the powder body. The die wall is rigid

so that it is a kinematic constraint and a radial stress

radial is developed. These two stresses allow forcomputing stress invariants p and q (or their related

quantities e Eq.(4) and m , Eq.(5)). Upper punch

displacement data permits volumetric strain

computation in turn.

1.1 Specimen size

First considerations come from specimen height to

cross section diameter relationship DH/ : the

larger DH/ , the more inhomogeneous and more

expensive the specimen. However, an DH/ toosmall would give a total punch displacement small

enough so that quite few experimental points could

be taken without interfering with punch travel

device accuracy figure. In addition, failure tests to

the powder body at different relative densities (e.g.

unconfined compression test, Brazilian disc test[12]) might impose a lower bound on DH/ ratioas well. This all adds to the need of making room

for the foil strain gage to capture actual data from

powder compaction. With a bore diameter of

12.5mm as starting point, and a maximum DH/ offour, pressure versus fractional porosity data from

references [11] were used to compute the required

punch displacements to seven figures of fill level

heights (from 30 mm to 116 mm), as set out in

Table 4.

Punch travel for different relative densities anddifferent fill level heights has been computed. As fill

level increases, so does punch travel. For a 30mm

figure, punch travel roughly ranges from 5mm to

10mm. Now it depends on the punch displacement

measuring device the decision whether this range is

large enough to take about 10 experimental points

without concern as to the measuring instrument

accuracy. For the present study a loose powder

cylindrical specimen of 30mm height and 12.5mm

diameter was deemed appropriate. It is worth to

recall, however, that obtaining compacts for failure

tests at different, intermediate densities shall require

fill level heights of at least 30mm.

1.2 Radial stress, hoop strain

radial is computed indirectly from the hoop (or

tangential) strain on die walls outer surface

.

Here the relationship( )radialf = is called

transduction equation, and is obtained as follows,

according to Figure 9.

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p i

a

b r

Figure 9. Geometric characterization of the closed die

cross section, for design purposes.

Considering the die wall as a thick walled cylinder,

hoop strain for plane strain condition is given by[13]:

( )[ ]zrvE

+=1

(21)

( ) += rz v (22)

At br= , where hoop strain is to be measured, there

is a free surface, so 0=r and one is left with:

( )[ ]211 vE

=

(23)

It may be found in reference [13] that for a non

rotating, thick walled cylinder:

+

=

2

2

22

2

1r

b

ab

pa i

(24)

Where ip stands for internal pressure. In the present

case this is precisely radial exerted by the loaded

specimen. Enforcing br= and substituting backgives:

( ) ( )

( )

radialradialabE

vaf

==22

22 12 (25)

Table 4.Necessary punch displacements (mm) to achieve densities near to theoretical, for typical metal powders [German,

1994], given several figures of fill level heights.

Material 1 -(Relative Density) P(Mpa) 30 45 60 75 90 100 116

0.35 0 0.00 0.00 0.00 0.00 0.00 0.00 0.00Bronze (Spherical)1

0.04 1000 9.69 14.53 19.38 24.22 29.06 32.29 37.46

0.22 100 0.00 0.00 0.00 0.00 0.00 0.00 0.00Copper (Spherical)2

0.054 520 5.26 7.90 10.53 13.16 15.79 17.55 20.36

0.28 150 0.00 0.00 0.00 0.00 0.00 0.00 0.00Iron (Sponge)2

0.12 610 5.45 8.18 10.91 13.64 16.36 18.18 21.09

0.4 150 0.00 0.00 0.00 0.00 0.00 0.00 0.00Stainless steel 2

0.16 770 8.57 12.86 17.14 21.43 25.71 28.57 33.14

Required punch displacements equal to zero indicate the onset of load application.

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Barrera et al.


And this is the transduction equation. Radial

stress within the specimen can be taken from Table

3. The resulting hoop microstrains for a 10mm die

wall thicknesses ( )ab are given in Table 5. Diewall is to be made of tool steel

( 3.0;200 == GPaE ).

Table 5. Radial stresses and hoop microstrains for a

compaction process.

Radial Stress (Mpa) Hoop microstrain

13.41 21.18

32.55 51.42

66.88 105.66

115.15 181.92

The hoop strains shown in Table 5 can be readily

measured by currently available foil strain gages, so

a die wall thickness of 10mm was selected to buildthe compaction die.

1.3 Axial stress, assemblage

The load cell beneath the powder specimen providesdata to asses a correction to upper punch pressure

with due to die wall-powder friction during

compaction. Both upper punch and a bottom plate

are rigid, whereas the load cell body is made of a

linear elastic material whose strain upon load

application is high enough to be captured by means

of an attached foil strain gage. References [3] show

that after total loading, load cell force is of up to

90% that from upper punch. This provides a clue on

design features.

From data in Table 4, upper punch pressure ranges

from about 100MPa the lowest to 1000MPa the

highest. This means a 90-900MPa range on the load

cell. At this point a decision was made in regards of

lower punch cross sectional area: the arrangement of

Figure 10 shows, in addition to the assemblage

description and picture, that this area is the same as

that of the specimen. This ensures that the lower

punch, which acts as a load cell, does not collapse at

high compaction pressures, and also that punchingforce line and bottom punch axis are adequately

aligned.

Metal plates onto the top punch and beneath the

bottom one were added to prevent the pressing

machine plates from denting.

TOP PLATE

UPPER PUNCH

DIE

METAL POWDER

BOTTOM PUNCH

STRAIN GAGE

LEAD PROVISION

HOOP STRAIN

GAGES

Figure 10.Instrumented closed die.

Using a tool steel bottom punch cell would convey a

5000strain under 1000MPa of pressure just overthe surface in contact with the powder. This hints a

limit in top punch applied stress should be

considered. Specifically, and in order to meet

reported punching pressures to attain high

compaction density in a wide variety of metal

powders of industrial interest, upper punch stress is

bounded to 800MPa, So that, at worst, a 3600strain (considering an up to 10% loss of punch

pressure on accounts of die wall-powder friction) is

obtained. A typical foil strain gage holds its

measuring features up until 4000.

1.4 Volumetric strain

Upper punch displacement is directly related tothe volume change of the specimen, and a strain can

be calculated thereof, Figure 11.

Ho

P

Figure 11.Data for volumetric strain calculation

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Here oH is the initial, fill level height. From data in

Table 4 may be of up to 10mm for oH of 30mm

(see discussion on specimen size determination).

This gives a volumetric strainV

Vv

= of about

33%. Thus, a logarithmic measure of strain must beconsidered, yielding:

=

==

o

o

o

f

vvH

H

V

V

V

dVd

lnln (26)

Relative density is frequently used to characterize

the material evolution throughout compaction, and

is computed as shown by Eq.(27).

( )( )

=

0

2

4

HD

mR

full

D

Bring m the mass of the specimen.(27)

1.5 Design tolerances

References from powder compaction tooling

designers [14] provided design tolerances and

material specifications (AISI D2 cold work tool

steel; hardness greater than of equal to 62HRC on

contact surfaces), which where also used to check

that elastic deformations under applied loads were

within design limits, and in particular, that radialstraining of the punches inside the die did not

interfere with the punching stress intended to

deform the powder body. Figure 12 shows

construction details.

55mm

MIN12.457mm

MAX12.484mm

MIN12.5mm

MAX12.527mm

75mm

17mm

32.5mm

3mm

15mm

MIN12.457mm

MAX12.484mm

25mm

0.010to 0.01545

Punch

Punch end detail

Bottom Punch

Compaction die

Upper Punch

Figure 12. Construction details of the closed die and

punches.

8 CONCLUDING REMARKS

In regards of the material model employed to solve

the closed die compaction problem in order to get a

figure of radial stress exerted by the material against

the die wall, it is worth remarking that it was a

conjunction of assumed linear elastic, small strain

behaviour, with a porosity dependent yield criterion

and a Von Mises plasticity recovering at high

relative densities. The validity of this model was

assumed on the reported elliptic shape of the yield

surface for metal foams, which were deemed an

appropriate example of porous bodies at low

densities; and that at full densities the material was a

ductile one, so that Von Mises yield surface proves

right.

The model was successfully solved for the close diecompaction case, and in particular, a maximum

value of radial stresses was the expected outcome. It

represents advancement in that, previously, trial and

test prototype building was the manner of finding

out radial stresses at the design stage of

instrumented dies.

Concerning the instrumented die construction three

hoop strain gages were attached to the outer surface

of the die. They are aimed at capturing a radial

stress variation along the specimen length on

accounts of sliding die wall friction. The setup was

used to study this problematic [15], whose resultsare expected to be published in the near future

Accuracy and precision of measurements can be

computed from basic standard data from strain

gages along with computations from Eq.(25) and

data in Figure 12 ( 3.0;200 == GPaE are takenfrom general references so their scatter is considered

zero); and also from pressing machine load cell

accuracy and cross section area of top punch,

bearing in mind its tolerances from Figure 12 as

well. Further refinements are aimed at specifying

uncertainty figures to alignment/misalignment of

hoop strains when attached onto the die surface.

9 REFERENCES

[1] Standard ASTM B331-85: Compressibility of

Metal Powders in Uniaxial Compaction, Vol.

02.05American Society for Testing and

Materials, 1990.

[2] U.W.S.(University of Wales Swansea), Civil andComputational Engineering Centre. MERLIN

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Barrera et al.


powder compaction software: users manual.

2007.

[3] Zavaliangos, A., Cunningham, J., Sinka, I.C. J.

of Pharm. Sci. 2004; 93 (8).

[4] Budynas, R. Advanced Strength and Applied

Stress Analysis. McGraw-Hill, 1999.

[5] Ashby, M.F. et al. Metal Foams: a Design

Guide. Butterworht-Heinemann, 2000.

[6] Heckel, R.W. Trans. Metall. Soc. AIME. 1961;

(221): 671.

[7] Deshpande, V.S., Fleck, N.A. J. Mech. Phys.

Solids. 1999; (48): 1253-1283.

[8] Green, R.J. International Journal of Mechanical

Sciences. 1972; (14): 215-224.

[9] Terzaghi, K. Soil Mechanics in Engineering

Practice. 2nd Edition. John Wiley & Sons,

1967.[10] Hoganas. Iron Powder Handbook. 1957; 1.

[11] German, R. Powder Metallurgy Science, 2nd

Edition. MPIF, Princeton, NJ., 1994.

[12] Stagg, R.G. Rock Mechanics in Engineering

Practice. John Wiley & Sons, 1968.

[13] Timoshenko, S. Theory of Elasticity, 3rdEdition. McGraw-Hill, 1970.

[14] Kunkel, R. Manufacturing Engineering &

Management. 1970; 240-244.

[15] Barrera, M. Construction of a True Compaction

Curve: Critical Review of the Closed DieCompaction Test for Powdered Materials.

PhD Thesis Cali (Colombia), Materials

Engineering School, Universidad del Valle,

2008.

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